Assignment_10

Assignment_10 - PDE 1 2 2 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛...

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ME 303 ADVANCED ENGINEERING MATHEMATICS Assignment 10 Question 1. Consider an insulated bar, length 5 = L , initially at 250°C, with Neumann and Robin BC’s applied for 0 > t . PDE: 2 2 x T t T = α BCs: 0 0 = = x x T 0 = + = = L x L x T x T ICs: () 250 0 , = x T Solve for () t x T , by separation of variables. a) Find numerical values for 1 λ , 2 , 3 . b) Demonstrate that orthogonality is valid by showing that 0 ) ( 2 1 = b a dx x p φ . c) Complete the solution for () t x T , . Question 2. A long, thick-walled, cylindrical tube has inner radius I R and outer radius O R . The tube is initially cold with uniform temperature C T . For 0 > t , the tube wall is heated by holding both inner and outer walls at a constant hot temperature H T . The temperature ( ) t r T , inside the tube wall is described by the diffusion equation in polar coordinates, plus the above IC and BCs; that is,
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Unformatted text preview: PDE 1 2 2 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ = ∂ ∂ r T r r T t T BCs H I T t T ,t) T(R = = ) , (R O IC C T T = ) , (r Use the subdivision idea discussed and demonstrated in class to get homogeneous BCs. Then, apply separation of variables to find ( ) t r T , . Your solution should consist of: (i) an infinite series for ( ) t r T , (ii) an equation which defines the eigenvalues (iii) definite integrals which define the coefficients of the series (do not evaluate the integrals) The solution should define the functional relationship ( ) C H O I T T R R t r f T , , , , ; , = Convective heat Transfer to a fluid at 0 [°C]...
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This note was uploaded on 09/16/2011 for the course ME 303 taught by Professor Serhiyyarusevych during the Spring '10 term at Waterloo.

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